I tried to shorten this down, but I want to quote this entire paragraph from early on in Hamlet's post, because it deserves to be repeated in its entirety:
The MVP of your raid group isn't the person at the top of the DPS meter; it's the person who hasn't died by standing in fire in 3 months. Doing elite DPS is rare, but doing quite good DPS is somewhat common. A player who does quite good DPS and dies regularly to avoidable mechanics is nothing special. A player who does quite good DPS and dies very rarely to avoidable mechanics is the best prize a guild recruiter can find in their applicant pool. Because a raid full of those players, even if none of them are at the top of worldwide log rankings, is going to succeed at what matters. They're going to win boss encounters.
I smiled at this analysis, because it reminded me of the philosophy at the center of the movie Moneyball, based on the book of the same title, which documents a specific evolution in the way baseball teams were formulated. The basic idea is that the most effective baseball team won't necessarily be composed of stars with the most home runs. It will be composed of the players who get on base the most. Because getting on base is ultimately what matters. A player on base is a player who can get around all the bases and score.
Similarly, this is what Hamlet points out: the player who very, very rarely dies to avoidable boss mechanics is a more valuable player than the one who tops the DPS meters, and a player who is both of those things is a true prize indeed. As my old raid leaders used to say: your DPS is zero when you're dead. Furthermore, I don't think this philosophy is unique to DPS players, though we often use them as examples. As a main spec healer, keeping myself alive is the best thing I can do for a raid, because I too am incapable of performing my duties while dead. If I'm so focused on healing other people that I forget to move out of the Falling Ash, then none of those people get any more of my heals.
So, the question remains: how to become a player who doesn't die in the fire? Hamlet goes through and offers some very specific advice for achieving this, and it is excellent. In addition, Hamlet's tips make clear some other things I've rarely seen articulated, for example, the fact that every single thing the boss does is probably not something you need to know about. Do you really care about cleave if you're not a tank or a tank healer? Does every single boss ability really need to have a sound trigger? So, first step then: identify the mechanics you absolutely must know about, and pare down your UI warnings to show only those things. Then, make sure that those warnings are extremely noticeable, so that you will never tune them out.
I could go on listing more examples of the advice in the post, but there's really no point in me continuing to repeat it. Just go read the whole thing already--seriously! The greatest thing about Hamlet's advice is that following it doesn't require anyone to be a serious theorycrafter, or to have the most bleeding edge equipment, or the World's Best Internet Connection. The single most important thing you can do to improve your raid performance is to become more situationally aware, and there are straightforward, relatively simple ways you can do that. It just may be the best thing you ever do for your raid team.